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**A new proof of desingularization over fields of characteristic zero.**
*(English)*
Zbl 1073.14021

In this paper, there is a new proof of the existence and a construction of a resolution of excellent schemes of finite type over a ground field of characteristic 0 and of principalization of an ideal of a smooth scheme.

This proof is based on the earlier works of Villamayor, Villamayor-Encinas and Bierstone-Milman, all working after Hironaka. The main point is as usual the “maximal contact”.

What is new ?

1. This proof is concise (14 pages).

2. The Hilbert-Samuel function is avoided.

The proof starts as usual by the construction of a “basic object” \((W_0, (J_0,b),E)\) where \(W_0\) is a smooth variety, \(J_0\subset{\mathcal O}_{W_0}\) an ideal, \(b\in \mathbb N\), \(E_0\) is a normal crossing divisor of \(W_0\). The proof is in two parts:

1- Resolve any basic object,

2- Prove that the resolution of basic objects leads to resolution of embedded schemes and to principalization of ideals.

In the case of desingularization, the main invariant used is the order of \(J_0\) restricted to \(W_0\) a smooth variety of maximal contact. The strata given by this order are not the Hilbert-Samuel strata. This is an improvement: these new strata can be computed easily, which is not the case for the H-S strata. Hence, this new algorithm is easier to implement.

A problem arises: the authors do not look at the strict transform of \({J_0}\), but at its weak transform \(t^{-b}{J_0}\) where div\((t)\) is the exceptional divisor of the blowing-up. This small modification with the usual proofs simplifies the redaction but, then, it is not clear at all that the algorithm is independent of the embedding: the weak transform \(t^{-b}{J_0}\) depends obviously on the embedding and on the choice of \(W_0\). This difficulty is easily overcome: the authors show that two different \(W_0\) and \(W'_0\) have same dimension and that you can find an étale covering \(\tilde W_0\) where the pull back of the basic objects on \(W_0\) and \(W'_0\) are the same, so the algorithms coincide on each pair \((W_0, \tilde W_0)\) and \((W'_0, \tilde W_0)\). This means that the authors’ construction has many properties of invariance that should be interpreted geometrically.

This proof is based on the earlier works of Villamayor, Villamayor-Encinas and Bierstone-Milman, all working after Hironaka. The main point is as usual the “maximal contact”.

What is new ?

1. This proof is concise (14 pages).

2. The Hilbert-Samuel function is avoided.

The proof starts as usual by the construction of a “basic object” \((W_0, (J_0,b),E)\) where \(W_0\) is a smooth variety, \(J_0\subset{\mathcal O}_{W_0}\) an ideal, \(b\in \mathbb N\), \(E_0\) is a normal crossing divisor of \(W_0\). The proof is in two parts:

1- Resolve any basic object,

2- Prove that the resolution of basic objects leads to resolution of embedded schemes and to principalization of ideals.

In the case of desingularization, the main invariant used is the order of \(J_0\) restricted to \(W_0\) a smooth variety of maximal contact. The strata given by this order are not the Hilbert-Samuel strata. This is an improvement: these new strata can be computed easily, which is not the case for the H-S strata. Hence, this new algorithm is easier to implement.

A problem arises: the authors do not look at the strict transform of \({J_0}\), but at its weak transform \(t^{-b}{J_0}\) where div\((t)\) is the exceptional divisor of the blowing-up. This small modification with the usual proofs simplifies the redaction but, then, it is not clear at all that the algorithm is independent of the embedding: the weak transform \(t^{-b}{J_0}\) depends obviously on the embedding and on the choice of \(W_0\). This difficulty is easily overcome: the authors show that two different \(W_0\) and \(W'_0\) have same dimension and that you can find an étale covering \(\tilde W_0\) where the pull back of the basic objects on \(W_0\) and \(W'_0\) are the same, so the algorithms coincide on each pair \((W_0, \tilde W_0)\) and \((W'_0, \tilde W_0)\). This means that the authors’ construction has many properties of invariance that should be interpreted geometrically.

Reviewer: Vincent Cossart (Versailles)

### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |

### Software:

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\textit{S. Encinas} and \textit{O. Villamayor}, Rev. Mat. Iberoam. 19, No. 2, 339--353 (2003; Zbl 1073.14021)

### References:

[1] | Abramovich, D. and de Jong, A. J.: Smoothness, semistability and toroidal geometry. J. Algebraic Geom. 6 (1997), no.4, 789-801. · Zbl 0906.14006 |

[2] | Abramovich, D. and Wang, J.: Equivariant resolution of singularities in characteristic 0. Math. Res. Lett. 4 (1997), no. 2-3, 427-433. · Zbl 0906.14005 · doi:10.4310/MRL.1997.v4.n3.a11 |

[3] | Aroca, J. M., Hironaka, H. and Vicente, J. L.: The theory of max- imal contact, Memorias Matemáticas Instituto Jorge Juan 29, Consejo Su- perior de Investigaciones Científicas, 1975. |

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